The arclength formula is
L=int_a^b sqrt(1+f'(x)) dx
First, find the derivative of the f(x)
d/dx(sqrt((x+3)(x/2-1))+5x)
d/dx(sqrt((x+3)(x/2-1)))+d/dx(5x)
d/dx(sqrt((x+3)(x/2-1)))+5
For the first term, let u=(x+3)(x/2-1)=1/2 x^2+1/2 x -3
(du)/dx =x+1/2
Using u-substitution, find d/dx(sqrt((x+3)(x/2-1)))
d/dx(sqrt(u))=1/(2sqrt(u)) (du)/dx
Replacing these values with x, gives
d/dx(sqrt(u))=(x+1/2)/(2sqrt(1/2 x^2+1/2 x -3))
Plug this result into the arc-length formula above
L=int_a^b sqrt(1+f'(x)) dx
L=int_6^7 sqrt(1+(x+1/2)/(2sqrt(1/2 x^2+1/2 x -3))) dx
This is a very difficult function to integrate, so using numeric methods is recommended. Using Simpson's Rule (with n=10 and Deltax=1//10), you could calculate
L~~(1//10)/3[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+cdots
cdots+4f(x_10)+f(x_11)]~~1.33